Properties

Label 4014.2.a.q
Level $4014$
Weight $2$
Character orbit 4014.a
Self dual yes
Analytic conductor $32.052$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4014,2,Mod(1,4014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4014.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4014.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.0519513713\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.10273.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 5x^{2} + x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1338)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + (\beta_1 - 1) q^{5} + ( - \beta_{3} - 1) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + (\beta_1 - 1) q^{5} + ( - \beta_{3} - 1) q^{7} + q^{8} + (\beta_1 - 1) q^{10} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{11} + (2 \beta_{3} - \beta_{2} - 2) q^{13} + ( - \beta_{3} - 1) q^{14} + q^{16} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{17} + (\beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{19} + (\beta_1 - 1) q^{20} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{22} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{23} + ( - \beta_{3} + \beta_{2} - 1) q^{25} + (2 \beta_{3} - \beta_{2} - 2) q^{26} + ( - \beta_{3} - 1) q^{28} + (\beta_{3} - 2 \beta_{2} - 3) q^{29} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{31} + q^{32} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{34} + (2 \beta_{3} - \beta_1) q^{35} + (\beta_{3} + 2 \beta_1 - 4) q^{37} + (\beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{38} + (\beta_1 - 1) q^{40} + (\beta_{3} + 3 \beta_{2} - 3) q^{41} + ( - \beta_{3} + \beta_{2} - 5) q^{43} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{44} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{46} + ( - 2 \beta_{2} + \beta_1 + 1) q^{47} + (2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{49} + ( - \beta_{3} + \beta_{2} - 1) q^{50} + (2 \beta_{3} - \beta_{2} - 2) q^{52} + (\beta_{3} - 2 \beta_{2} + 3 \beta_1 - 3) q^{53} + ( - \beta_{2} - 3 \beta_1 - 2) q^{55} + ( - \beta_{3} - 1) q^{56} + (\beta_{3} - 2 \beta_{2} - 3) q^{58} + ( - 4 \beta_{3} + \beta_1 + 2) q^{59} + ( - \beta_{3} - 2 \beta_1 - 1) q^{61} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{62} + q^{64} + ( - 3 \beta_{3} - 3 \beta_1 + 3) q^{65} + ( - \beta_1 - 5) q^{67} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{68} + (2 \beta_{3} - \beta_1) q^{70} + (2 \beta_{3} + \beta_{2} - 3 \beta_1) q^{71} + ( - 2 \beta_{3} - 3 \beta_{2} - 1) q^{73} + (\beta_{3} + 2 \beta_1 - 4) q^{74} + (\beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{76} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{77} + (\beta_{3} - 2 \beta_{2} + 4 \beta_1 - 10) q^{79} + (\beta_1 - 1) q^{80} + (\beta_{3} + 3 \beta_{2} - 3) q^{82} + (3 \beta_{3} - \beta_{2} + 2) q^{83} + (3 \beta_{3} - \beta_{2} + 3 \beta_1 - 5) q^{85} + ( - \beta_{3} + \beta_{2} - 5) q^{86} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{88} + (3 \beta_{3} + 4 \beta_1 - 5) q^{89} + (3 \beta_1 - 6) q^{91} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{92} + ( - 2 \beta_{2} + \beta_1 + 1) q^{94} + ( - 3 \beta_{3} - \beta_{2} + 1) q^{95} + (\beta_{3} + 3 \beta_1 - 9) q^{97} + (2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} - 2 q^{5} - 5 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{4} - 2 q^{5} - 5 q^{7} + 4 q^{8} - 2 q^{10} - 4 q^{11} - 5 q^{13} - 5 q^{14} + 4 q^{16} + 2 q^{17} - 7 q^{19} - 2 q^{20} - 4 q^{22} + 4 q^{23} - 6 q^{25} - 5 q^{26} - 5 q^{28} - 9 q^{29} - q^{31} + 4 q^{32} + 2 q^{34} - 11 q^{37} - 7 q^{38} - 2 q^{40} - 14 q^{41} - 22 q^{43} - 4 q^{44} + 4 q^{46} + 8 q^{47} - 7 q^{49} - 6 q^{50} - 5 q^{52} - 3 q^{53} - 13 q^{55} - 5 q^{56} - 9 q^{58} + 6 q^{59} - 9 q^{61} - q^{62} + 4 q^{64} + 3 q^{65} - 22 q^{67} + 2 q^{68} - 5 q^{71} - 3 q^{73} - 11 q^{74} - 7 q^{76} - 2 q^{77} - 29 q^{79} - 2 q^{80} - 14 q^{82} + 12 q^{83} - 10 q^{85} - 22 q^{86} - 4 q^{88} - 9 q^{89} - 18 q^{91} + 4 q^{92} + 8 q^{94} + 2 q^{95} - 29 q^{97} - 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 5x^{2} + x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 2\nu^{2} - 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu^{2} - 2\nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 2\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 3\beta_{2} + 8\beta _1 + 6 \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.38266
−0.641043
0.673533
3.35017
1.00000 0 1.00000 −2.38266 0 1.61326 1.00000 0 −2.38266
1.2 1.00000 0 1.00000 −1.64104 0 −3.78585 1.00000 0 −1.64104
1.3 1.00000 0 1.00000 −0.326467 0 −1.59754 1.00000 0 −0.326467
1.4 1.00000 0 1.00000 2.35017 0 −1.22988 1.00000 0 2.35017
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(223\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4014.2.a.q 4
3.b odd 2 1 1338.2.a.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1338.2.a.g 4 3.b odd 2 1
4014.2.a.q 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

\( T_{5}^{4} + 2T_{5}^{3} - 5T_{5}^{2} - 11T_{5} - 3 \) Copy content Toggle raw display
\( T_{7}^{4} + 5T_{7}^{3} + 2T_{7}^{2} - 13T_{7} - 12 \) Copy content Toggle raw display
\( T_{11}^{4} + 4T_{11}^{3} - 13T_{11}^{2} - 9T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} - 5 T^{2} - 11 T - 3 \) Copy content Toggle raw display
$7$ \( T^{4} + 5 T^{3} + 2 T^{2} - 13 T - 12 \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} - 13 T^{2} - 9 T + 16 \) Copy content Toggle raw display
$13$ \( T^{4} + 5 T^{3} - 18 T^{2} - 45 T + 108 \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} - 43 T^{2} + 75 T - 2 \) Copy content Toggle raw display
$19$ \( T^{4} + 7 T^{3} - 28 T^{2} - 189 T - 104 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} - 57 T^{2} + 69 T + 514 \) Copy content Toggle raw display
$29$ \( T^{4} + 9 T^{3} - 157 T - 312 \) Copy content Toggle raw display
$31$ \( T^{4} + T^{3} - 46 T^{2} + 25 T + 122 \) Copy content Toggle raw display
$37$ \( T^{4} + 11 T^{3} + 7 T^{2} - 92 T - 141 \) Copy content Toggle raw display
$41$ \( T^{4} + 14 T^{3} - 25 T^{2} + \cdots + 1088 \) Copy content Toggle raw display
$43$ \( T^{4} + 22 T^{3} + 171 T^{2} + \cdots + 648 \) Copy content Toggle raw display
$47$ \( T^{4} - 8 T^{3} - 7 T^{2} + 43 T - 27 \) Copy content Toggle raw display
$53$ \( T^{4} + 3 T^{3} - 66 T^{2} - 97 T + 942 \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} - 101 T^{2} + \cdots - 136 \) Copy content Toggle raw display
$61$ \( T^{4} + 9 T^{3} - 8 T^{2} - 163 T - 256 \) Copy content Toggle raw display
$67$ \( T^{4} + 22 T^{3} + 175 T^{2} + \cdots + 747 \) Copy content Toggle raw display
$71$ \( T^{4} + 5 T^{3} - 69 T^{2} - 441 T - 584 \) Copy content Toggle raw display
$73$ \( T^{4} + 3 T^{3} - 133 T^{2} + \cdots + 4003 \) Copy content Toggle raw display
$79$ \( T^{4} + 29 T^{3} + 207 T^{2} - 2 T - 3 \) Copy content Toggle raw display
$83$ \( T^{4} - 12 T^{3} - 5 T^{2} + 323 T - 289 \) Copy content Toggle raw display
$89$ \( T^{4} + 9 T^{3} - 170 T^{2} + 43 T + 68 \) Copy content Toggle raw display
$97$ \( T^{4} + 29 T^{3} + 242 T^{2} + \cdots - 1048 \) Copy content Toggle raw display
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